The Fourier Transform of the Complex Exponential

The complex exponential function is common in applied mathematics. The basic form is written in Equation [1]:       [1] The complex exponential is actually a complex sinusoidal function. Recall Euler's identity:       [2] Recall from the previous page on the dirac-delta impulse that the Fourier Transform of the shifted impulse is the complex exponential:       [3] If we know the above is true, then the inverse Fourier Transform of the complex exponential must be the impulse:       [4] What we are interested in is the Fourier Transform of the complex exponential in equation [1], or:       [5] Using the second line of Equation [4], equation [5] can be quickly solved:       [6] The last equal sign equation follows directly from equation [4]. Hence, the Fourier Transform of the complex exponential given in equation [1] is the shifted impulse in the frequency domain. This should also make intuitive sense: since the Fourier Transform decomposes a waveform into its individual frequency components, and since g(t) is a single frequency component (see equation [2]), then the Fourier Transform should be zero everywhere except where f=a, where it has infinite energy.